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This module builds the tools necessary for the frequency analysis of brain waves as recording by an electroencephalograph. We proceed from the Pythagorean Theorem to sine waves, the trapezoid rule and finally to Fourier decomposition.

Table of contents

0. Background

1. Sine and Cosine Waves

2. Trapezoid Rule for estimating area

3. Fourier Method for decomposing signals

4. Spectrogram application to analyzing brain waves

Background: brain waves and the eeg

Signals are sent through the brain using both chemical and electrical means. The synchronized electrical activity of individual neurons adds up to something big enough to detect on from outside the head. To measure it, we use a set of electrical nodes called an electroencephalogam (EEG). The measured activity reflects different states of the brain which in turn tell us something about the mindset of the person. Our goal in this module is to decompose an EEG signal into its different frequencies, which is intuitively the most meaningful piece of information.

Sine and cosine waves

Brainwaves have complex shapes that are not easily interpreted. In order to study these waves, we need to develop some mathematical tools that will tell us about different waves. To outline, we begin by talking about pure (sine or cosine) waves, then move to the trapezoid rule for estimating area under a curve. Next, we develop Fourier analysis for picking out the frequencies in a jumbled signal, and finally use these tools to create spectrograms, which allow us to track different frequencies over time.

Sine waves

The sine wave is a mathematical function. It describes many physical phenomena, including sound waves and oscillation. It looks just like a wave. MATLAB uses the sin function to make sin waves. For example, to make Figure 1, we use the code:

>>t = 0:.01:1;>>y = sin(2*pi*t);>>plot(t,y);

The sine wave is defined by the lengths and angles of a triangle. Run sincirc.m (copied below) to see how the sine and cosine values relate to the angle ϕ of the triangle. As you can see, if ϕ is the angle of a right triangle with hypotenuse 1 (illustrated by the circle) , sin ( ϕ ) is the height of the triangle and cos ( ϕ ) is the base of it:

A sin wave

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The relation of sine and cosine to a triangle and unit circle

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Two illustrations of the sine function.
% sincirc.m %% sincirc.m illustrates the relation of the sin and cosine waves to the circle. %define parametersNturns = 2; steps_per_turn = 9;step_inc = 2*pi/steps_per_turn; %set up points for circlecirc_x = cos(0:.01:2*pi); circ_y = sin(0:.01:2*pi);axis equal %loop over triangles with different anglesfor n = 1:Nturns * steps_per_turn; phi = n * step_inc + pi/4;%plot circle, then triangle, then text plot(circ_x, circ_y);axis([-1 1 -1 1] * 1.5);line([0 cos(phi)], [0 sin(phi)]); line([1 1]* cos(phi), [0 sin(phi)]);line([0 cos(phi)], [0 0]); text(cos(phi)/2 , -.1*sign(sin(phi)),'cos(\varphi)')text(cos(phi) + .1*(sign(cos(phi))-.5), sin(phi)/2, 'sin(\varphi)') text(cos(phi)*.2, sin(phi)*.1,'\varphi');pause(.5); end

Characteristics of the sine wave

The sin wave has three primary characteristics:

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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